Each of the above items are discussed in the next subsections.
We will use the following parity and periodicity properties for elliptic functions [1]}:
\(&\begin{array}{l}\mathrm {sn}(-u)=-\mathrm {sn}{\;u},\qquad \qquad \mathrm {cn}(-u)=\mathrm {cn}{\;u},\qquad \qquad \;\mathrm {dn}(-u)=\mathrm {dn}{\;u}.\end{array}\vspace{5.69054pt}\\&\begin{array}{l}\mathrm {sn}(u+2K)=-\mathrm {sn}{\;u},\qquad \mathrm {cn}(u+2K)=-\mathrm {cn}{\;u},\qquad \mathrm {dn}(u+2K)=\mathrm {dn}{\;u}.\end{array}\) 
